Topological alternatives and structural modeling of

Research paper

Struct Multidisc Optim 25, 1–11 (2003) DOI 10.1007/s00158-003-0325-4

Topological alternatives and structural modeling of infinite grillages on elastic foundations M.B. Fuchs, M. Ryvkin and E. Grosu

Abstract Several infinite spatially periodic grillages on elastic supports, subjected to lateral static loading, are studied. The topologies of the repeating patterns of the grillages that are considered herein are the only possibilities for dividing a plane into regular polygons. These are grillages are composed of triangular, square, or hexagonal cells. A general scheme for the analysis of arbitrary periodic infinite grillages on elastic supports is presented. The compliance and robustness, which is understood as the response of a system when one of its elements is absent, are calculated analytically for the case of discrete elastic spring supports and of a Winkler elastic foundation. A parametric study is carried out in order to compare the three topologies. The results are discussed and concluding remarks are presented. Key words

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1 Introduction Periodically arranged systems of beams or grillages are a standard tool for bridging spans or reinforcing elastic plates. Another known use of grillages is supporting or protecting structures when they rest on some kind of elastic support. This paper is dedicated to studying the fundamental stiffness and reliability properties for this type of grid of beams. We consider in particular the three typical patterns that divide the plane into regular polygons: equilateral triangles, rectangles (squares), and hexagons. Received: December 2001 Revised manuscript received: August 2002 2003 Published online:  Springer-Verlag 2003 M.B. Fuchsu , M. Ryvkin, E. Grosu Department of Solid Mechanics, Materials and Systems, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University. 69978 Tel Aviv. Israel e-mail: [email protected], http://www.eng.tau.ac.il/~fuchs

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Two cases are considered: lattices elastically supported at junctions and lattices on an elastic foundation. The prime objectives is to obtain the minimum compliance or minimum stress levels for a load applied at a typical node using a fixed amount of material. In the case of discrete elastic supports, the comparison between the different configurations is carried out using an equal number of supports per unit covered area under a constant volume constraint or using identical beams. In the case of a continuous elastic foundation, the comparison is for an equal volume of material (beams and supports) per unit covered surface. The comparison of these topological alternatives hinges evidently on the analysis of infinite repetitive structures under arbitrary loading. In some sense this problem is similar to the dynamic response of infinite grillages with damping under harmonic excitation. Such infinite grillages with rectangular cells have been intensively studied by Langley (1996) and Langley et al. (1997). Herein, static analysis under arbitrary loading is considered in the context of the Representative Cell method (Ryvkin and Nuller 1997). Using the discrete Fourier transform (DFT) the infinite problem is reduced to a problem defined over a single cell. When based on a finite element implementation (Moses et al. 2001) the technique leads to ‘equilibrium’ equations in terms of the nodal displacements and force transforms in conjunction with Born–Von Karman-type boundary conditions. After solving the complex-valued system of equations the real displacements in any cell of the infinite assembly is obtained by the inverse transform. This method was successfully applied to the optimal design of infinite trusses of 1D translational symmetry (Ryvkin et al. 1999) and to the topological design of structures of 1D translational and of cyclic symmetry (Moses et al. 2001). The generalization to 2D translational symmetry is straightforward, as exemplified in the present work. After comparing the stiffness and stress levels of these grillages, a robustness test is performed so as to study the structural response of the three basic patterns for cases in which one of their components is missing. When the structure loses a member, although the arrangement is no longer periodic, we use the Structural Variation method (Majid et al. 1978, Fuchs and Hakim 1996) to

2 analyze the infinite grid. This technique is part of a family of methods, such as the Virtual Distortion method (Kolakowsi and Holnicki-Szulc 1998) and the Sherman– Morrison–Woodbury formulas (Akg˝ un et al. 2001), which base the analysis of the modified structure on the analysis of the original (symmetric in the present context) structure under the applied and additional loading conditions. The final displacements are then a combination of the various displacement fields. Interestingly, these techniques were developed in the realm of structural optimization with the purpose of alleviating the numerical burden of multiple reanalysis. In the present case, however, structural variations and related methods seem to be the only approaches for dealing with infinite structures in which the repetitive modules are identical but for a few missing or modified elements. We will show in the following that, as a rule, the triangular mesh performs best, closely followed by the squares, with the hexagonal grid in last position. Although this intuitive result could have been predicted, it nevertheless needed to be established. For instance, previous research on fracture analysis of cellular materials (Chen et al. 1998) had the fracture toughness in a different sequence (highest for hexagonal or triangular, then square). In addition the quantitative differences between the considered topologies are also of interest and cannot be determined without a thorough analysis. The next section is dedicated to the analysis of infinite grids on elastic supports. This is followed by a comparative study of the three basic grid topologies with 2D symmetry, which concentrates on the compliance and reliability. Numerical results are then reported and, prior to the conclusions, robustness is investigated.

2 Infinite periodic grillages on elastic supports Consider an infinite regular beam grillage that rests on elastic supports and is subjected to arbitrary transverse nodal loading. As noted by Renton (1966), any regular plane pattern can be subdivided into identical parallelogram modules (Fig. 1). We do not specify the grillage topology at the present time, thus emphasizing the generality of the approach. The vertices of the parallelograms do not necessarily represent nodes of the grillage. The 2D translational symmetry of the grids can be defined by the translation vectors (b1 , b2 ) that constitute the sides of the parallelograms. Employing these vectors as a basis, one can see that the location of a parallelogram mesh node is determined by a vector index k = {k1 , k2 }, where the integers k1 and k2 define the radius vector b1 k1 + b2 k2 . This notation for describing a symmetric mesh is well known in the theory of lattices (Gutkowski 1974). It will be recognized that a vector basis related to the given periodic plane mesh is not unique, and we consider one of several possibilities. The interested reader can find additional information on the subject in Weyl (1952). As indicated, since we are examining a structure of 2D symmetry, the Representative Cell method (Ryvkin and Nuller 1997) can be used for the analysis. This method enables the analysis of an arbitrarily loaded mechanical system possessing translational symmetry to be reduced to the analysis of the repetitive module: the representative cell. In accordance with this approach we view the infinite grillage on elastic supports as a juxtaposition of identical cells (parallelograms in Fig. 2a). Each cell is assumed to carry the vector index k of its vertex for which the num-

k (k 1,k2 ) b2 (0,0)

b 1k 1 + b 2k 2 b1

Fig. 1 Node numbering in a symmetric grid

3 bers k1 , k2 are individually minimum, i.e., the index of the lower left corner. Besides the corner nodes at the parallelogram vertices, the cell may include boundary nodes, symmetrically located on periodic boundaries, as well as inner nodes. The corner nodes belong to four neighboring cells, the boundary nodes belong to two neighboring cells, and the inner nodes are, of course, related to a single cell. Herein we will consider only grids with triangular, square, and hexagonal patterns. As will be shown below, these topologies can be construed as subsets of the module shown in Fig. 2b without boundary nodes and only one inner node. The case of 1D translational symmetry (in which corner nodes cannot exist) with several inner and boundary nodes is considered in detail in (Ryvkin et al. 1999). The node notation is identical in every cell. As illustrated in Fig. 2b the representative cell has four corner nodes and an inner one. It is important to delineate the generic pattern within the representative cell. This pattern will reconstitute the entire structure when translated along k1 b1 + k2 b2 . In the figure we note that it includes the generic nodes 00 (black dots), the beams (full lines), and, in the case of grids on point supports, also the springs under the generic nodes. For grids on continuous supports the discrete springs are absent and the beams are assumed to be continuously supported (not shown in the figure). The dashed lines merely complement the boundary of the cell. Clearly, in each cell k only generic nodes can carry the external nodal loading of cell k. Beams located at the interface between cells will be related to the cell with larger number k1 or k2 (see Fig. 2a and b). A Cartesian coordinate system xyz with the z axis perpendicular to the grillage plane is introduced in each cell as shown in Fig. 2b. The vector of the generalized nodal displacements for node (αβ) of cell k is (k) (k) defined as uαβ = {w, θx , θy }αβ and has three components

in accordance with the three degrees of freedom, namely the displacement in the z direction and the rotations with respect to the x and y axes, respectively. The vector of the corresponding generalized external forces acting on (k) (k) a node includes the external forces pαβ = {P, Mxp , Myp }αβ and the contact forces applied by the neighboring cells (k) (k) fαβ = {F, Mxf , Myf }αβ . Using the above definitions one can formulate the problem for the infinite grillage in the following manner. The stress–strain state in a cell number k satisfies the equilibrium equation written in terms of 5 × 3 = 15 degrees of freedom Ku(k) = p(k)+ f (k) , k = {k1 , k2 } , k1 , k2 = 0, ±1, ±2 , ... , (1) where K is the stiffness matrix of the cell and u(k) = {u00 , u−− , u+− , u−+ , u++ }(k)

(2)

is the vector of nodal displacements. In the vector of external forces p(k) = {p00 , p−− , 0, 0, 0}(k) ,

(3)

only the first two components are non-zero since only they are associated with the grillage nodes. On the other hand, in the vector of resultants applied to cell k by its neighbors f (k) = {0, f−− , f+− , f−+ , f++ }(k) ,

(4)

the first component is zero because there are no contact forces at an inner node.

Fig. 2 (a) Exploded view of cell k1 k2 and adjoining cells of plane grid with 2D symmetry. (b) Typical cell: The generic (repeating) module is the black nodes and full lines. The dotted lines complete the cell boundary. The white nodes are on the cell boundary but belong to neighboring cells

4

(k +1,k2 )

,

(5)

and E is the unit diagonal 3 × 3 matrix. We will denote the matrix transposed and conjugate to B by B H (Hermitian). It is instructive to note that the equilibrium condition (14) yields the congruent relation

(k ,k2 +1)

,

(6)

BH f = 0 .

(7)

Pre-multiplying (10) by matrix BH and using (16)–(19), one obtains the reduced system with respect to the displacements of the generic nodes of the representative cell

(8)

K u = p ,

The conditions at the cell boundaries (corner points) include the continuity conditions for the displacements (k ,k2 )

1 = u−−

(k ,k2 )

1 = u−−

(k ,k2 )

1 = u−−

1 u+−

1 u−+

u++1

(k +1,k2 +1)

,

and the equilibrium conditions (k ,k2 )

f++1

(k +1,k2 )

+ f−+1

(k ,k2 +1)

+ f+−1

(k +1,k2 +1)

+ f−−1

=0.

Application of the two-dimensional discrete Fourier transform (DFT) g=

∞


∞


g (k1 ,k2 ) eiϕ1 k1 +iϕ2 k2

(9)

(19)

(20)

where p = BH p ,

(21)

K = BH KB .

(22)

k1 =−∞ k2 =−∞

to the infinite system of equations (1), (5)–(8) leads to the problem for the representative cell Ku = p + f ,

(10)

u+− = γ1−1 u−− ,

(11)

u−+ = γ2−1 u−− ,

(12)

u++ = γ1−1 γ2−1 u−− ,

(13)

f++ + γ1−1 f−+ + γ2−1f+− + γ1−1 γ2−1 f−− = 0 ,

(14)

where γj = e

iϕj

,

j = 1, 2 .

(15)

The force and displacement transforms in this problem are complex-valued and depend upon the transform parameters ϕ1 , ϕ2 . For brevity we omit hereafter the word “transform” when referring to quantities related to the problem for the representative cell, that is, we will use “displacements” instead of “displacement transforms”, etc. The solution of (10)–(14) is straightforward. Using (11)–(13) we can express the displacements of all five nodes of the cell by means of the displacements of the generic nodes u−− and u00 : u = Bu ,

(16)

where u = {u00 , u−− } ,   E 0             0 E       −1 γ1 E B= 0 ,        0 γ2−1 E           −1 −1  0 γ1 γ2 E

Mead et al. (1988) have obtained a similar condensed matrix K in the dynamic problem of a 2D periodic array of elements. Having obtained vector u one can find the remaining nodal displacements in the cell from (16) and hence the internal forces and moments in the beams. Recall that these quantities are in fact transforms of actual displacements and forces. To find the real displacements and forces in any cell one has to apply the inverse DFT. For instance, the displacement of node (−−) in cell k is (k) u−−

1 = 2 4π

π π

u−− e−iϕ1 k1 −iϕ2 k2 dϕ1 dϕ2 .

(23)

−π −π

We will assume that all the beams of the cell have uniform bending stiffness EI, where E is the Young modulus and I is the second moment of the area of the element. The stiffness of the discrete springs is k and the stiffness per unit length of the foundation is c. The torsional stiffness is taken as negligible small. In the case of a grid on point supports the global stiffness matrix K in (1) is assembled from the classical uniform free beam element matrices and their spring stiffnesses. For the analysis of the second case of infinite grids on continuous elastic supports (Winkler foundation), we need to use the element stiffness matrix of a beam on an elastic foundation. In contrast with the free beam, the solution of the 4-th order differential equation of a beam on an elastic foundation is expressed not in terms of polynomials but by a combination of exponential and trigonometric functions (see, for example, Eisenberger and Yankelevsky (1985)).

(17) 3 Triangular, square and hexagonal grids

(18)

Using the technique described in the previous section, one can now determine the response of any regular infinite grillage to arbitrary loading. As mentioned we will consider here the only three topological alternatives of

5 regular polygons that can cover a plane. These are equilateral triangles, squares, and equilateral hexagons. (subscripts t for triangle, s for square, and h for hexagon will henceforth denote the grillage topology.) A perusal of the representative cells of these grillages, as illustrated in Fig. 3, indicates that their respective topologies can be construed as subsets of the topology given in Fig. 2b. For the triangular grillage the inner node (00) is absent, u ≡ u−− . The element has four nodes (−−), (+−), (+−), (++), one of them generic (−−). Keeping in mind that every node has one translational and two rotational degrees of freedom, the stiffness matrix of the three-beam representative cell in (1) is of 12-th order, and matrix Bt (18), relating the displacements of the four nodes to those of the generic node, is of order 12 × 3. The square grillage is the simplest one. Its representative cell consists of only three nodes (−−), (+−), (+−) connected by two beams. Hence, the stiffness matrix has order 9, vector u is the same as in the previous case, and matrix Bt , relating the non-generic nodes of the cell with the generic node (−−), is of order 9 × 3. Finally, the representative cell for the hexagonal grillage is the only one that has an inner node. The element has four nodes (00), (−−), (−+), (+−), the first two being the generic nodes. Therefore, the order of the stiffness matrix is 12, vector u has two components, and Bh is an 12 × 6 matrix:

Bt =

Bh =

 E       γ −1 E 1

      

 γ2−1 E          −1 −1   γ1 γ2 E   E 0           0 E 

;

γ1−1 E     −1  γ2 E

.

 0      0

  E       −1 Bs = γ1 E ;     −1   γ2 E

and continuous reactions), which are calculated in accordance with (22), are given in the Appendix. It is recognized that the non-dimensional parameters

kL3 c and λ = 4 , (25) β= EI 4EI where L is the typical length of the beams in each case, play an important role in the analysis of the structures. They characterize the balance between the stiffness of the beams of the grillage and the discrete and continuous supports, respectively. Next, one has to derive the transformed loading vector p in accordance with (9) and (21), and eventually solve the system (20). A unique non-singular solution always exists, since the rigid body motion of the representative cell is eliminated thanks to the elastic support and boundary conditions. Finally, by applying the inverse DFT (23) one can compute the structural response in any cell k of the grillage. Before proceeding we need to define a basis of comparison for the three topologies in order to put them, as much as possible, on the same footing. The grillages on continuous supports call for a simple solution. Indeed, a natural constraint that comes to mind is to impose a fixed weight of grillage material per unit covered area. Since we assume that the cross-section of the beams for all grillages is identical, the tributary area Sj per unit beam length in all three cases is Sj =

Lj π − ωj tan , 2 2

j = t, s, h ,

(26)

where ω (see Fig. 2b) is the angle between the translation vectors b1 and b2 . Since for the considered meshes (see Fig. 3) ωt = 2π/3, ωs = π/2 and ωh = π/6 the weight constraint implies the following ratios between the beam lengths for the three grids: √ √ Lt : Ls : Lh = 3 : 1 : 1/ 3 or 1.732 : 1 : 0.577 . (27) (24)

Note that the B matrices in (24) are subsets of (18). The condensed matrices Kmn for the three topological alternatives and for the two types of elastic supports (point

The layouts of the corresponding grillages are drawn to scale in Fig. 3. It will be noted that when viewed from afar the three topologies will, in this case, exhibit the same grey scale.

Fig. 3 The three grids (to scale, different beams) in the case of a discrete foundation with constant weight per unit covered area (Case 2b). One beam is missing

6 The comparison of grillages resting on point spring supports requires a different equivalence criterion. Indeed, the grillages shown in Fig. 3 have different numbers of nodal points per unit covered area. Consequently, since the springs are identical, this will result in different re-

sultant stiffness of the supports per unit area. In order to keep this stiffness constant we stipulate that the number of nodal points per unit area for different grillages is the same. This will result in other ratios between the beam lengths:

Fig. 4 The three grids (to scale) in the case of a continuous elastic foundation. The respective topologies of the repetitive modules are also shown

Table 1 Basis of comparison of the triangular, square and hexagonal topologies. The size of the grids was adjusted to maintain in Case 1 the same density of material and in Cases 2 the same density of springs. c: stiffness per unit length, k: stiffness of the springs, L: typical beam length, W : weight per unit covered area, I: second moment of area of the beams. Case 2a: same beams, Case 2b: same weight density Case

Support

Comparison

1 2a 2b

Continuous (c) Discrete (k)

Material Springs

Lt : Ls : Lh

Wt : Ws : Wh

It : Is : Ih

1.732 : 1.0 : 0.577 1.075 : 1.0 : 0.877 1.075 : 1.0 : 0.877

1.0 : 1.0 : 1.0 1.615 : 1.0 : 0.568 1.0 : 1.0 : 1.0

1.0 : 1.0 : 1.0 1.0 : 1.0 : 1.0 0.619 : 1.0 : 1.52

7  Lt : Ls : Lh =

2 2 √ :1: √ 3 3 3

or 1.075 : 1 : 0.877 . (28)

To complete the conditions of the comparison it is plausible to make one of the following two additional assumptions: (a) The cross-sections of the beams used in the different grillages are identical. (b) The weight of the grillages per unit area is the same. Hence, in the first case the specific weight for different grillage layouts does not remain constant: Wt : Ws : Wh =

1 1 1 : : = 1.615 : 1 : 0.658 , St Ss Sh

(29)

where S is the tributary area of every beam element in each case. Maintaining a constant weight per unit area in the second case, Wt = Ws = Wh

(constant specific weight) ,

(30)

was achieved by varying the width of the beams while keeping the height constant. Consequently, the second moments of the area of the beams with lengths (28) are related as It : Is : Ih = 0.619 : 1 : 1.52 .

(31)

Such grids (drawn to scale) are depicted in Fig. 4. (Please ignore the missing beam at this point.) For the sake of convenience the terms for the comparison are summarized in Table 1.

4 Numerical results The main criterion for testing the structures is the compliance of the structure. Maximum bending moments

(and maximum stresses, where required) are also computed. In a subsequent section the robustness of the grids is checked by studying the perturbations to displacements and internal forces that are experienced by removing an element adjacent to the loaded node. We will assume that the loading is a single point force P applied to node k = {0, 0}: (k)

p−− = {P δ0k1 δ0k2 , 0, 0} ,

(32)

where δ0k1 , δ0k2 are Kronecker deltas. The force transform is thus simply p−− = {P, 0, 0}. The expression for the compliance, which is understood as the vertical displacement of the loaded node, has the general form w≡

(0) w−−

1 = 2 4π

π π w−− dϕ1 dϕ2 .

(33)

−π −π

The explicit expressions for the displacement transforms, being rather cumbersome, are not exhibited. It may be said from the outset that on all accounts the triangular grids carry the day. As a rule, they are stiffer, develop lower stresses, and as will be shown in the next section, they are more robust than their square and hexagonal counterparts. Some typical results corroborate this statement. Consider, for instance, grids with identical beam cross-sections either on an elastic foundation (Case 1 in Table 1) or on discrete springs (Case 2a in Table 1). The continuous case is depicted in the graph to the left in Fig. 5, in which the normalized displacement of the loaded node w ˆ = w/ P/c and the non-dimensional maximum moment (adjacent to the load) M/P Ls are given as a function of the non-dimensional parameter αc , αc = (λLs )−1 ,

1

αk = (EIs /kL3s ) 4 . (34)

Here M is the bending moment in the beams under the loaded node, which is obviously the largest bending moment in the grillage. The parameter αc characterizes the

Fig. 5 Normalized maximum displacement and moment for grids on the elastic foundation (left) and discrete supports (right) as a function of α, a measure of the ratio of the beam stiffness EI to the support stiffness, c and k, respectively

8

Fig. 6 Normalized maximum displacements, moments, and stresses for grids on discrete supports, with the same weight per unit covered area as a function of αk , a measure of the ratio of the beam stiffness EI to the spring stiffness k

ratio of the stiffness of the beams of the grillage over the stiffness of the elastic foundation. The descending curves depict the compliance and the ascending curves describe the bending moments. It can be seen that the triangular mesh is the stiffest, the square mesh is more compliant, and the hexagonal one is most flexible. Similarly, the bending moment increases when passing from the triangle to the square to the hexagon. Low values of αc correspond to small values of the beam stiffnesses compared with the foundation stiffness. In the limiting case, when αc → 0, the displacements become very large while maintaining a constant ratio for the compliances, w ˆt : w ˆs : w ˆh = 1/6 : 1/4 : 1/3. This is expected since the load is carried by 6, 4, and 3 beams (in fact, elastic foundations) for the triangle, square, and hexagonal grillages, respectively. The analysis shows that increasing the beam stiffnesses results in a rapid decrease of the compliance, while the ratio between the results for the different topological alternatives remains approximately the same. The behavior of the maximum bending moment is in agreement with the results for the displacements. It is found that the highest bending moment occurs in the hexagonal grillage, the lowest is in the triangular one, with the square grillage in the middle. The ratio between the bending moments in the entire range is found to be similar to that of the displacements, and the general trend is a monotonic increase with increasing beam stiffness. We can now turn our attention to the grids on discrete springs presented in the graph to the right of Fig. 5. The ordinate is here αk , defined in (34), which is a measure of the ratio of the grid stiffness to the spring stiffness. Contrary to the case of a continuous elastic foundation, when the beams stiffness vanishes (αk → 0), the normalized compliances tend in all cases to the same value (the compliance of the spring under the load), and the bending moment is zero. Indeed, the supporting action, in the absence of grillages, is provided only by the spring under the loaded node. With increasing beam stiffness, as was previously found, the compliance decreases and the

maximum bending moment increases. Here, the triangular grillage is also found to be the stiffest with the lowest bending moments, and the hexagonal design is the most compliant with the largest bending moments. When opting for constant weight in the discrete case (line 2b in Table 1), as a basis for comparison, we notice rather surprisingly that the compliances of the triangular and square grillages are almost identical (see Fig. 6, left). The compliance of the hexagonal grillage is, as usual, larger but the difference is moderate. On the other hand, the bending moments maintain their typical discrepancies. However, since the moments of inertia vary between the grids (see line 2b in Table 1), it is also of interest to examine the maximum bending stresses (Fig. 6, right) rather than the bending moments. As shown, the maximum bending stresses for the triangular and square grillages almost coincide, but are always less then those of the hexagonal grillage.

5 Robustness The robustness of the structures was tested by removing a beam adjacent to the loaded node and investigating the perturbation to the displacements and the internal forces (Fig. 3). Some care must be exercised in the analysis since the structure is no longer symmetric. Apparently this precludes the use of the Representative Cell method, which is applicable only to infinite periodic structures. We therefore employed the Structural Variation method (Majid 1978), which belongs to a family of techniques in which the response of the modified (non-symmetric in our case) structure is based on the analysis of the original (symmetric) one under the applied and additional loadings. For this purpose we also utilize the concept of unimodal elements, by which a beam element is decomposed into its constituent unimodal moment and shear elements (Fuchs 1991, 1997). The moment element carries the aver-

9 age bending moment m in pure bending and the shear element carries the shear force s and the associated differential bending moments in ‘pure’ shear. Let us assume that one beam, that is, two unimodal elements, is to be removed. We would like to analyze the modified structure on the basis of results for the original one. These results include the analysis of the original structure under the external loads, under unit and opposite moments applied to the nodes connected to the beam that is to be removed, and under equal and opposite shears (and balancing moments) applied at the same nodes. The displacements in these three cases will be denoted u, um , and us (see Fig. 7 for the case of the square grid). As a rule, superscripts m or s mean ‘as a result of unit moments or unit shears applied to the nodes of the beam to be removed.’ Quantities without superscripts relate to the external loads. ˆ of the modified One can show that the displacement u structure under general external loads is  

m  m m s s ˆ ˆ ˆ = u + mˆ u u u + sˆ u = u+ u , (35) s where m and s are, respectively, the moment and shear in the beam of the original structure under the external ˆ m and u ˆ s are, respectively, the displacements loads, and u of the modified structure under unit moment and shear. ˆ m and u ˆ s are not known at this stage. Now, Note that u (35) is valid for any loading, in particular for a unit moment and shear applied to the modified structure:   m

m  m m m ˆ ˆs u ˆ =u + u , u sm  

m  ms s s s ˆ ˆ ˆ =u + u u u , ss or

ˆm u

 1 − mm  ˆs u −ms



−sm 1−s

s

= um

 us ,

(36)

which, by post-multiplying both sides by the inverse of ˆ m and u ˆ s: the coefficients matrix, leads to u  −1  m  1 − mm −sm

m s s ˆ ˆ = u u u u . (37) −ms 1 − ss

p

1

1

Introducing (37) into (35) provides all the displacements for the modified structure under general loads in terms of analysis results of the original one:  −1   1 − mm −sm m

m  us ˆ = u+ u . (38) u s s −m 1−s s It can be seen that by using this method the modified non-symmetric structure can be analyzed in its pristine symmetric configuration, thus allowing the DFT approach. It is noteworthy that the Structural Variation method and related techniques were developed in the realm of structural optimization in order to abate the numerical burden of the multiple reanalysis inherent to minimization algorithms (Kirsch 1993, Haftka et al. 1990). In this field the techniques are very instrumental but are not absolutely required. It seems, however, that in the present context of slightly modified but otherwise symmetric structures, the Structural Variation and related methods are indispensable for dealing with the predicament of non-symmetry. Numerical results show that the triangular grid is also the preferred topology here. Case 2b (discrete spring supports, same specific weight), for instance, is depicted in Fig. 4. The data in fine print give an indication of the propagation of the perturbation inside the structures. The top numbers are the nominal bending moments normalized with respect to the maximum absolute moment in each grid. The bottom values (in parentheses) are the modified values obtained after removing the critical beam. By omitting the beam the deflection under the loaded node relative to the original one increases as shown in Table 2. The triangular grid experiences an additional deflection of less than 10%, followed by 30% for the squares. The hexagonal grid is the most vulnerable.

Table 2 Perturbation of the maximum displacements and moments for grids on discrete springs (same weight density, Case 2b), due to a missing beam contiguous to the loaded node. Nominal values are 1.00 Grid

Displacement

Triangle Square Hexagon

1.09 1.32 3.51

1 L/2

u

um

Moment 1.55 1.62 0.00

L/2 1

us

Fig. 7 The three loading conditions to mimic the removal of an element: external forces (left), and unit pure moment and ‘pure’ shear applied to the element to be removed (L is the element length)

10 As noted in the last column of Table 2 the near bending moments of the two remaining beams adjacent to the loaded node are zero and the displacement increases more than threefold. For all practical purposes the hexagonal grid ceases to function when it loses a critical beam.

6 Conclusions The topologies of infinite repetitive grids of uniform beams on elastic supports with triangular, square, and hexagonal cells were compared. The grids rest on either an elastic foundation or discrete springs under the nodes. In the former case the individual mesh sizes were adjusted so as to maintain a fixed length of beams per unit covered area. For the discrete springs the mesh sizes were determined by imposing a fixed number of nodes per unit covered area. The analysis was performed by the Representative Cell method by means of the discrete Fourier transform. This allowed a single module to be analyzed, albeit in transformed (complex-valued) nodal displacements. As a rule the triangular grids were stiffer, had lower stress levels, and were more robust. Next in line were the square grids, followed by the hexagonal ones. In one case (discrete springs with fixed weight density) the triangular and square grids gave almost identical displacements and stress levels for the entire range of spring and beam stiffnesses. The main purpose of this paper was to quantify the structural response of grids composed of identical regular polygons. Although infinite structures are a theoretical concept, their structural behavior can often replicate the response of their finite counterparts when end effects are negligible. For a concentrated nodal force we have clearly established the dominance of the triangular solution. Although easier to implement than the triangular grid, the putative square solution is less effective. The hexagonal grids should be treated with much caution. Their seemingly dense structure and aesthetic appeal are misleading. From the strict point of view of structural behavior, they should be dismissed. It is noteworthy that the condensed matrices appearing in the Appendix make it possible to generalize the results to arbitrary loading. Moreover, the technique described herein is equally valid for periodic elastic grillages with more complex cell geometries.

References Akg˝ un, M.A.; Garcelon, J.H.; Haftka, R.T. 2001: Fast exact linear and non-linear structural reanalysis and the Sherman– Morrison–Woodbury formulas. Int. J. Numer. Methods Eng. 50(7), 1587–1606 Chen, J.Y.; Huang, Y.; Ortiz, M. 2001: Fracture analysis of cellular materials: A strain gradient model. J. Mech. Phys. Solids 46(5), 789–828

Dean, D.L. 1976: Discrete Field Analysis of Structural Systems. Courses and Lectures No. 203. Udine: CISM Eisenberger, M.; Yankelevsky D.Z. 1985: Exact stiffness matrix for beams on elastic foundation. Comput. Struct. 21(6), 1355–1359 Fuchs, M.B. 1991: Unimodal beam elements. Int. J. Solids Struct. 27(5), 533–545 Fuchs, M.B. 1997: Unimodal formulation of the analysis and design problems for framed structures. Comput. Struct. 63(4), 739–747 Fuchs, M.B.; Hakim, S. 1996: Improved multivariate reanalysis of structures based on the structural variation method. Mech. Struct. Mach. 24(1), 51–70 Gutkowski, W. 1974: Mechanical problems of elastic lattice structures. In: Kuchemann, D. (ed.) Progress in Aerospace Science, Vol. 15. Oxford: Pergamon Press Haftka, R.T.; G¨ urdal, Z.; Kamar, M.P. 1990: Elements of Structural Optimization, 2nd revised edn. Dordrecht: Kluwer Academic Publishers Kirsch, U. 1993: Structural Optimization: Fundamentals and Applications. Berlin: Springer–Verlag Kolakowski, P; Holnicki-Szulc, J. 1998: Sensitivity analysis of truss structures (virtual distortion method approach). Int. J. Numer. Methods Eng. 43(6), 1085–1108 Langley, R.S. 1966: The response of two-dimensional periodic structures to point harmonic forcing. J. Sound Vibration 197(4), 447–469 Langley, R.S.; Bardell, N.S.; Ruivo, H.M. 1997: The response of two-dimensional periodic structures to harmonic point loading: a theoretical and experimental study of a beam grillage. J. Sound Vibration 207(4), 521–535 Majid, K.I.; Saka, M.P.; Celik, T. 1978: The theorems of structural variation generalized for rigidly jointed frames. Proc. Inst. Civ. Eng. 51(2), 839–856 Mead, D.J.; Zhu, D.C.; Bardell, N.S. 1988: Free vibration of an orthogonally stiffened flat plate. J. Sound Vibration 127(1), 19–48 Moses, E.; Ryvkin, M.; Fuchs, M.B. 2001: A FE methodology for the static analysis of infinite periodic structures under general loading. Comput. Mech. 27, 369–377 Renton, J. 1966: On the analysis of triangular mesh grillages. Int. J. Solids Struct. 2, 307–318 Ryvkin, M.; Fuchs, M.B.; Nuller, B. 1999: Optimal design of infinite repetitive structures. Struct. Optim. 18(2/3), 202–209 Ryvkin, M.; Nuller, B. 1997: Solution of quasi-periodic fracture problems by the representative cell method. Comput. Mech. 20, 145–149 Wah, T.; Calcote, L.R. 1970: Structural Analysis by Finite Difference Calculus. New York: Van Nostrand Weyl, H. 1952: Symmetry. Princeton: Princeton University Press

11 

Appendix: Stiffness matrices




Ksc

The following condensed matrices for triangular (subscript t), square (s), and hexagonal (h) cells are calculated in accordance with (22). Condensed matrices for point support (subscript p): K
tp =     

EI × L3

β + 72 − 24(c1 + c12 + c2 )

conjugate

K
sp

  β + 48 − 24(c1 + c2 ) −12Lis1 −12Lis2   EI   4L2 (2 + c1 ) 0 = 3    L  2 conjugate 4L (2 + c2 )

K
hp = 

 √ 6 3Li(s12 + s2 )  √  3L2 (c12 − c2 )  L2 (12 + 4c1 + c12 + c2 )  3L2 (4 + c12 + c2 ) −6Li(2s1 + s12 − s2 )

  + 4λ2 h+ s − h (c1 + c2 )

−4iλs1 hs

−4iλs2 hs

− h− s + 2c1 h

0

conjugate

K
hc =

+ ) −12(1 + γ12

β + 36

0

6Lγ2−

0

−6Lγ1−

0

− 6Lγ21

8L2

2L2

4L2

8L2

−2L2 γ2 8L2

− −6Lγ12

     2 2L γ1    −4L2    +   2L2 γ12  0

8L2



where γj = exp(iϕj )

± γij = γi ± γj−1

sj = sin(ϕj ) cj = cos(ϕj ) c12 = cos(ϕ1 + ϕ2 ) s12 = sin(ϕ1 + ϕ2 )

d = 2(sinh2 α − sin2 α) d1 = EIλ

d1 × K
tc = d   √  8λ2 3h+ − 2h+ × 8iλhs × −8 3iλhs × s    (c1 + c12 + c2 )  (s2 − s12 − 2s1 ) (s12 + s2 )     √     − − − 2 3h (c12 − c2 )  2 h (4c1 + c12 + c2 ) + 3hs     −    conjugate 6 h (c12 + c2 ) + h− s

     

 + − 4λ2 h+ (1 + γ12 ) 0 4λ2 hs γ2− 0 −4λ2 γ12 6λ2 h+ s       − 6λ2 h+ −4λhs γ1− 0 −4λhs γ21 0   s       − − − − 2h hs 2h γ1  2hs       − −    2h− −2h γ −h 2 s s      − − +  conjugate  2h 2h γ s 12     2h− s

α = λL Condensed matrices for Winkler foundation (subscript c):

− h− s + 2c2 h



d1 × d

γj± = 1 ± γj−1

EI × L3

β + 36             conjugate 

 d1   =  d  

h± = sin α cosh α ± cos α sinh α hs = sin α sinh α h± s = sinh 2α ± sin 2α h± c = cosh 2α ± cos 2α